Introduction to Transparent Bayesian Data Analysis (chi2023-course)
Abhraneel Sarma and Fumeng Yang
03 April, 2023
Introduction
In this document, we will outline the Bayesian analogs of the statistical analyses described in lecture 1 (Github code).
Load packages
library(dplyr)
library(tibble)
library(purrr)
library(tidyr)
library(forcats)
library(gtools)
library(patchwork)
library(broom)
library(broom.mixed)
library(modelr)
library(brms)
library(tidybayes)
library(ggdist)
library(bayesplot)
library(ggplot2)
library(knitr)
BRM_BACKEND <- ifelse(require("cmdstanr"), 'cmdstanr', 'rstan')Dataset
The following dataset is from experiment 2 of “How Relevant are Incidental Power Poses for HCI?” (Jansen & Hornbæk, 2018). Study participants were asked to either make an expansive posture or a constrictive posture before performing a task. The experiment investigated whether posture could potentially have an effect on risk taking behavior.
First, we load the data.
pose_df = readr::read_csv("data/poses_data.csv", show_col_types = FALSE) %>%
mutate(condition) %>%
group_by(participant)The data has been aggregated for each participant: -
condition = expansive indicates expansive posture, and
condition = constrictive indicates constrictive posture -
The dependent variable is change which indicates the
percentage change in risk-taking behavior. Thus, it is a continuous
variable.
For the purposes of this demo, we are only concerned with these two variables. We can ignore the other variables for now.
head(pose_df %>% select(participant, condition, change))| participant | condition | change |
|---|---|---|
| 1 | expansive | 3.362832 |
| 2 | constrictive | 29.147982 |
| 3 | expansive | 25.409836 |
| 4 | constrictive | 54.069767 |
| 5 | expansive | -36.644592 |
| 6 | constrictive | 29.756098 |
Intuition of Bayesian Statistics
The Bayesian t-test (BEST) assumes that the data in the two conditions arises from two separate t-distributions. In the following section, we will describe the process for one of the conditions in the experiment.
We will use the \(Normal(\mu = 20, \sigma = 20)\) as the prior distribution.
First, we define some functions for manual calculation of the posterior normal distribution:
sigma_post = function(sigma_prior, sigma, n = 1) {
sqrt(1 / (1 / (sigma_prior^2) + n / (sigma^2)))
}
mu_post = function(mu_prior, sigma_prior, mu, sigma, n = 1) {
tau = sigma_post(sigma_prior, sigma, n)
(tau^2 / sigma_prior^2)*mu_prior + (n * tau^2 / sigma^2)*mu
}d.p2 = tibble(
group = c("prior", "expansive", "constrictive", "posterior"),
mu = c(20, 32.82, 31.61, mu_post(20, 20, 32.82, 7.52)),
sd = c(20, 7.52, 7.06, sigma_post(20, 7.52))
) %>%
mutate(
cutoff_group = list(c(1:7)),
cutoff = list(c(0, 15, 25, 30, 32.82, 40, 100))
) %>%
unnest(c(cutoff_group, cutoff)) %>%
mutate(
cutoff = if_else(group == 'prior', 100, cutoff),
x = map(cutoff, ~ seq(from = -40, ., by = 0.1)),
y = pmap(list(x, mu, sd), ~ dnorm(..1, ..2, ..3))
)In the plot below, we show the raw data distribution for the two conditions:
p1 = pose_df %>%
ggplot() +
geom_point(aes(x = change, y = condition, colour = condition),
position = position_jitter(height = 0.1), alpha = 0.7) +
scale_color_theme() +
labs(y = "Condition") +
theme(
legend.position = "none",
axis.line.y = element_blank(),
axis.ticks.y = element_blank(),
axis.title.y = element_blank()
) +
scale_x_continuous(limits = c(-150, 250), breaks = seq(-150, 250, by = 50))
p2.blank = tibble(y = c("expansive", "constrictive"), x = 0) %>%
ggplot(aes(x, y)) +
scale_color_theme() +
theme_density
cowplot::plot_grid(p2.blank, p1, nrow = 2)
First, we define a function to help us plot different cutoff groups.
plot_preliminary <- function(data, group_num, group_name){
data %>%
filter(cutoff_group == group_num & group %in% group_name) %>%
unnest(c(x, y)) %>%
ggplot(aes(x, y)) +
#geom_line(aes(color = group), size = 1) +
# density
geom_area(aes(fill = group, color = group), position = "identity", linewidth = 1, alpha = 0.3) +
scale_x_continuous(limits = c(-40, 100)) +
scale_y_continuous(limits = c(0, 0.1)) +
scale_color_theme() +
theme_density
}cowplot::plot_grid(
plot_preliminary(d.p2, 0, NULL),
p1, nrow = 2)
Next, we plot the prior density:
cowplot::plot_grid(
plot_preliminary(d.p2, 1, 'prior'),
p1, nrow = 2)
Then we describe step by step, how the likelihood is computed:
cowplot::plot_grid(
plot_preliminary(d.p2, 1, 'expansive'),
p1, nrow = 2)
cowplot::plot_grid(
plot_preliminary(d.p2, 2, 'expansive'),
p1, nrow = 2)
cowplot::plot_grid(
plot_preliminary(d.p2, 3, 'expansive'),
p1, nrow = 2)
cowplot::plot_grid(
plot_preliminary(d.p2, 5, 'expansive'),
p1, nrow = 2)
cowplot::plot_grid(
plot_preliminary(d.p2, 6, 'expansive'),
p1, nrow = 2)
cowplot::plot_grid(
plot_preliminary(d.p2, 7, 'expansive'),
p1, nrow = 2)
We want to compute the posterior, which is the product of the prior and likelihood:
cowplot::plot_grid(
plot_preliminary(d.p2, 7, c('prior', 'expansive', 'posterior')),
p1, nrow = 2)
Of course, if you have gganimate and magick
working on your computer, you can try the following code to get a gif!
Here, we are not going to show the code.
Fumeng: the content below needs better documentation….
Model 1: Equal standard deviations
Step 1: model specification
Fumeng: I hope to keep this example simple. Is there a simple way to justify the choice of \(\nu\)?
Here we use a mathematical expression for the model above. Fumeng: need explanations here…
\[ \begin{align} y_{i} &\sim \mathrm{Student\_t}(\mu, \sigma_{0}, \nu_{0}) \\ \mu &= \beta_{0} + \beta_{1} * x_i \\ \sigma_{0} &\sim \mathrm{HalfNormal}(0, 10) \\ \beta_{0} &\sim \mathrm{?} \\ \beta_{1} &\sim \mathrm{Normal}(0,2) \\ \nu_{0} &\sim \mathrm{?} \\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\} \end{align} \]
We translate this thought into brms formula using the
function bf.
model.1.formula <- bf(
# we think change is affected by different conditions.
change ~ condition,
# to tell brms which response distribution to use
family = student()
)Now we can use get_prior() to inspect the available
priors and formula. As what we learned, we have priors for
tibble(get_prior(model.1.formula, pose_df))| prior | class | coef | group | resp | dpar | nlpar | lb | ub | source |
|---|---|---|---|---|---|---|---|---|---|
| b | default | ||||||||
| b | conditionexpansive | default | |||||||
| student_t(3, 22.6, 38.8) | Intercept | default | |||||||
| gamma(2, 0.1) | nu | 1 | default | ||||||
| student_t(3, 0, 38.8) | sigma | 0 | default |
prior checks
Look at the default priors
cowplot::plot_grid(
tibble(x = qstudent_t(ppoints(n = 500), df = 3, mu = 22.6, sigma = 38.8)) %>%
ggplot() +
geom_density(aes(x = x), fill = theme_yellow, color = NA) +
ggtitle('student_t(3, 22.6, 38.8) for Intercept') +
coord_cartesian(xlim = c(-300, 300),expand = c(0)),
tibble(x = qgamma(ppoints(n = 500), shape = 2, rate = .1)) %>%
ggplot() +
geom_density(aes(x = x), fill = theme_yellow, color = NA) +
ggtitle('Gamma(2, 0.1) for nu') +
coord_cartesian(xlim = c(1, 100), expand = 0),
tibble(x = qstudent_t(ppoints(n = 500), df = 3, mu = 0, sigma = 38.8)) %>%
ggplot() +
geom_density(aes(x = x), fill = theme_yellow, color = NA) +
ggtitle('student_t(3, 0, 38.8) for sigma') +
coord_cartesian(xlim = c(0, 400),expand = c(0)),
ncol = 3)
prior check for default priors
model.1.checks_default <- brm(
model.1.formula,
data = pose_df,
family = student_t(),
prior = c(
prior(normal(0, 3), class = 'b')
),
sample_prior = 'only',
#backend = BRM_BACKEND,
file = 'rds/model.1.checks_default.rds',
file_refit = 'on_change'
)plot
model.1.defaultpriorsamples <-
model.1.checks_default %>%
epred_draws(tibble(condition = c('expansive', 'constrictive')))head(model.1.defaultpriorsamples)| condition | .row | .chain | .iteration | .draw | .epred |
|---|---|---|---|---|---|
| expansive | 1 | NA | NA | 1 | 19.425813 |
| expansive | 1 | NA | NA | 2 | 38.452771 |
| expansive | 1 | NA | NA | 3 | -23.169166 |
| expansive | 1 | NA | NA | 4 | 48.733025 |
| expansive | 1 | NA | NA | 5 | 72.826955 |
| expansive | 1 | NA | NA | 6 | -2.607648 |
model.1.defaultpriorsamples %>%
ggplot(aes(x = .epred, group = condition)) +
geom_density(alpha = .5, color = NA, adjust = 2, fill = theme_yellow) +
theme_density_x +
ggtitle('Checks default priors of model.1')
Our priors
cowplot::plot_grid(
tibble(x = qnorm(ppoints(n = 1000), mean = 0, sd = 2)) %>%
ggplot() +
geom_density(aes(x = x), fill = theme_yellow, color = NA) +
ggtitle('Normal(0, 2) for b & Intercept') +
coord_cartesian(xlim = c(-8, 8), expand = 0),
tibble(x = qnorm(ppoints(n = 1000), mean = 0, sd = 10)) %>%
ggplot() +
geom_density(aes(x = x), fill = theme_yellow, color = NA) +
ggtitle('HalfNormal(0, 10) for sigma') +
coord_cartesian(xlim = c(0, 50), expand = c(0)),
ncol = 2)
model.1.checks <- brm(
model.1.formula,
data = pose_df,
family = student_t(),
prior = c(
prior(normal(0, 2), class = 'b'),
prior(normal(0, 10), class = 'sigma', lb = 0)
),
sample_prior = 'only',
#backend = BRM_BACKEND,
file = 'rds/model.1.checks.rds',
file_refit = 'on_change'
)plot
model.1.priorsamples <-
model.1.checks %>%
epred_draws(tibble(condition = c('expansive', 'constrictive')))Compare two sets of priors
cowplot::plot_grid(
model.1.defaultpriorsamples %>%
ggplot(aes(x = .epred, group = condition)) +
geom_density(alpha = .5, color = NA, adjust = 2, fill = theme_yellow) +
theme_density_x +
#coord_cartesian(xlim = c(-800, 800)) +
ggtitle('Checks default priors of model.1')
,
model.1.priorsamples %>%
ggplot(aes(x = .epred, group = condition)) +
geom_density(alpha = .5, color = NA, adjust = 2, fill = theme_yellow) +
theme_density_x +
#coord_cartesian(xlim = c(-800, 800)) +
ggtitle('Checks our priors of model.1')
,
ncol = 1)
Step 2: model fitting
model.1 <- brm(
model.1.formula,
data = pose_df,
family = student_t(),
prior = c(
prior(normal(0, 2), class = 'b'),
prior(normal(0, 10), class = 'sigma', lb = 0)
),
backend = BRM_BACKEND,
file = 'rds/model.1.rds'
)Step 3: check posteriors
aspect 1: mcmc traces
color_scheme_set("teal")
mcmc_trace(model.1, facet_args = list(ncol = 4))
aspect 2: model metrics
Rhat, ESS, etc here
summary(model.1)## Family: student
## Links: mu = identity; sigma = identity; nu = identity
## Formula: change ~ condition
## Data: pose_df (Number of observations: 80)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 22.84 4.49 14.50 31.77 1.00 2686 2092
## conditionexpansive -0.26 1.97 -4.10 3.62 1.00 2726 2987
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 28.35 4.13 20.76 36.92 1.00 2180 2520
## nu 3.68 2.35 1.57 9.04 1.00 2172 2158
##
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).aspect 3: visually
Fumeng: do we need this?
model.1.predictions <-
model.1 %>%
predicted_draws(tibble(condition = c('expansive', 'constrictive')))head(model.1.predictions)| condition | .row | .chain | .iteration | .draw | .prediction |
|---|---|---|---|---|---|
| expansive | 1 | NA | NA | 1 | -6.461254 |
| expansive | 1 | NA | NA | 2 | 54.432844 |
| expansive | 1 | NA | NA | 3 | -39.512069 |
| expansive | 1 | NA | NA | 4 | -182.802314 |
| expansive | 1 | NA | NA | 5 | -18.709933 |
| expansive | 1 | NA | NA | 6 | -10.418057 |
plot_predictions <- function(model, df = NULL, title = ''){
if(is.null(df))
df = tibble(condition = c('expansive', 'constrictive'),
participant = c(-1, -1))
model %>%
predicted_draws(df,
seed = 1234,
ndraws = NULL,
allow_new_levels = TRUE,
sample_new_levels = 'uncertainty') %>%
ggplot(aes(x = .prediction, fill = condition)) +
geom_density(alpha = .5, size = 1, adjust = 2, color = NA) +
scale_color_theme() +
theme_density_x +
scale_y_continuous(breaks = 0, labels = 'constrictive') +
scale_x_continuous(breaks = seq(-150, 250, by = 50)) +
coord_cartesian(xlim = c(-150, 250)) +
ggtitle(paste0('Posterior predictions of ', deparse(substitute(model))))
}cowplot::plot_grid(plot_predictions(model.1),
p1 +
scale_x_continuous(breaks = seq(-150, 250, by = 100)) +
coord_cartesian(xlim = c(-150, 250)),
nrow = 2)
model.1.posteriors <-
model.1 %>%
epred_draws(tibble(condition = c('expansive', 'constrictive')))plot_posteriors <- function(model, df = NULL, title = ''){
if(is.null(df))
df = tibble(condition = c('expansive', 'constrictive'))
model %>%
epred_draws(df,
# ignoring random effects if there is any
#seed = 1234,
ndraws = NULL,
re_formula = NA) %>%
ggplot(aes(x = .epred, fill = condition)) +
geom_density(alpha = .5, size = 1, adjust = 2, color = NA) +
scale_color_theme() +
theme_density_x +
scale_y_continuous(breaks = 0, labels = 'constrictive') +
scale_x_continuous(limits = c(-150, 250), breaks = seq(-150, 250, by = 50)) +
ggtitle(paste0('Posterior means of ', deparse(substitute(model))))
}cowplot::plot_grid(plot_posteriors(model.1), p1, nrow = 2)
Model 2: the BEST test model
Step 1: model specification
This model is the BEST test model as described by Kruschke in the paper Bayesian estimation supersedes the t-test. In this model, \(\beta\) indicates the mean difference in the outcome variable between the two groups (in this case, the percent change in the BART scores). We fit different priors on \(\beta\) and set different weights on these priors to obtain our posterior estimate.
\[ \begin{align} y_{i} &\sim \mathrm{T}(\nu, \mu, \sigma) \\ \mu &= \beta_{0} + \beta_{1} * x_i \\ \sigma &= \sigma_{a} + \sigma_{b}*x_i \\ \beta_{1} &\sim \mathrm{Normal}(\mu_{0}, \sigma_{0}) \\ \sigma_a, \sigma_b &\sim \mathrm{Cauchy}(0, 2) \\ \nu &\sim \mathrm{exp}(1/30)\\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\} \end{align} \]
model.2.formula <- bf(# we think change is affected by different conditions.
change ~ condition,
sigma ~ condition,
# to tell brms which response distribution to use
family = student())tibble(get_prior(model.2.formula, pose_df))| prior | class | coef | group | resp | dpar | nlpar | lb | ub | source |
|---|---|---|---|---|---|---|---|---|---|
| b | default | ||||||||
| b | conditionexpansive | default | |||||||
| student_t(3, 22.6, 38.8) | Intercept | default | |||||||
| gamma(2, 0.1) | nu | 1 | default | ||||||
| b | sigma | default | |||||||
| b | conditionexpansive | sigma | default | ||||||
| student_t(3, 0, 2.5) | Intercept | sigma | default |
prior check
Fumeng: not sure qcauchy has the correct paramaterization
cowplot::plot_grid(
tibble(x = qnorm(ppoints(n = 1000), mean = 0, sd = 2)) %>%
ggplot() +
geom_density(aes(x = x), fill = theme_yellow, color = NA) +
ggtitle('Normal(0 ,2) for Intercept') +
coord_cartesian(expand = c(0)),
tibble(x = qexp(ppoints(n = 1000), rate = 0.0333)) %>%
ggplot() +
geom_density(aes(x = x), fill = theme_yellow, color = NA) +
ggtitle('exponential(0.0333) for nu') +
coord_cartesian(expand = 0),
tibble(x = qcauchy(ppoints(n = 1000), location = 0, scale = 2)) %>%
ggplot() +
geom_density(aes(x = x), fill = theme_yellow, color = NA) +
ggtitle('cauchy(0, 2) for sigma') +
coord_cartesian(expand = c(0)),
ncol = 3)
model.2.priorchecks <- brm(
model.2.formula,
data = pose_df,
family = student_t(),
prior = c(
prior(normal(0, 2), class = 'b'),
prior(cauchy(0, 2), class = 'b', dpar = 'sigma'),
prior(exponential(0.0333), class = 'nu')
),
sample_prior = 'only',
#backend = BRM_BACKEND,
file = 'rds/model.2.priorchecks.rds',
file_refit = 'on_change'
)model.2.priorchecks %>%
epred_draws(tibble(condition = c('expansive', 'constrictive'))) %>%
ggplot(aes(x = .epred, group = condition)) +
geom_density(alpha = .5, color = NA, adjust = 2, fill = theme_yellow) +
theme_density_x +
ggtitle('Checks priors of model.2')
Step 2: model fiting
model.2 <- brm(
model.2.formula,
data = pose_df,
family = student_t(),
prior = c(
prior(normal(0, 2), class = 'b'),
prior(cauchy(0, 2), class = 'b', dpar = 'sigma'),
prior(exponential(0.0333), class = 'nu')
),
backend = BRM_BACKEND,
file = 'rds/model.2.rds',
file_refit = 'on_change'
)Step 3: posterior checks
summary(model.2)## Family: student
## Links: mu = identity; sigma = log; nu = identity
## Formula: change ~ condition
## sigma ~ condition
## Data: pose_df (Number of observations: 80)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept 24.48 4.69 15.61 34.16 1.00 2536
## sigma_Intercept 3.41 0.20 3.00 3.80 1.00 2430
## conditionexpansive -0.14 1.97 -3.94 3.67 1.00 4222
## sigma_conditionexpansive 0.15 0.22 -0.29 0.57 1.00 3731
## Tail_ESS
## Intercept 2052
## sigma_Intercept 2331
## conditionexpansive 2869
## sigma_conditionexpansive 2860
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## nu 6.16 9.42 1.73 24.18 1.00 1848 1380
##
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).wrap_plots(plot_predictions(model.2) + scale_x_continuous(limits = c(-200,250), expand = c(0,0), breaks = seq(-150, 250, by = 50))
,
p1 + scale_x_continuous(limits = c(-200,250), expand = c(0,0), breaks = seq(-150, 250, by = 50)) ,
nrow = 2)
# draws.model.3 =
wrap_plots(
plot_posteriors(model.2),
p1,
nrow = 2)
model.2.posteriors <- model.2 %>%
epred_draws(tibble(condition = c('expansive', 'constrictive')),
#seed = 1234,
ndraws = NULL,
re_formula = NA) Model 3: the BEST model with better intercepts
Step 1: model specification
\[ \begin{align} y_{i} &\sim \mathrm{T}(\nu, \mu, \sigma) \\ \mu &= \beta_{i,j} + \beta_{1} * x_i \\ \sigma &= \sigma_{a} + \sigma_{b}*x_i \\ \beta_{1} &\sim \mathrm{Normal}(0, 2) \\ \sigma_a, \sigma_b &\sim \mathrm{HalfNormal}(0, 10) \\ \nu &\sim \mathrm{exp}(1/30)\\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\}\\ j & \in \{1, ..., \mathrm{N}\} \end{align} \]
model.3.formula <- bf(# we think change is affected by different conditions.
change ~ condition + (1|participant),
sigma ~ condition,
# to tell brms which response distribution to use
family = student())tibble(get_prior(model.3.formula, pose_df))| prior | class | coef | group | resp | dpar | nlpar | lb | ub | source |
|---|---|---|---|---|---|---|---|---|---|
| b | default | ||||||||
| b | conditionexpansive | default | |||||||
| student_t(3, 22.6, 38.8) | Intercept | default | |||||||
| gamma(2, 0.1) | nu | 1 | default | ||||||
| student_t(3, 0, 38.8) | sd | 0 | default | ||||||
| sd | participant | default | |||||||
| sd | Intercept | participant | default | ||||||
| b | sigma | default | |||||||
| b | conditionexpansive | sigma | default | ||||||
| student_t(3, 0, 2.5) | Intercept | sigma | default |
Step 2: model fiting
model.3 <- brm(
model.3.formula,
data = pose_df,
family = student_t(),
prior = c(
prior(normal(0, 2), class = 'b'),
prior(normal(0, 2), class = 'b', dpar = 'sigma'),
prior(normal(0, 2), class = 'sd', group = 'participant', lb = 0),
prior(exponential(0.0333), class = 'nu')
),
#backend = BRM_BACKEND,
file = 'rds/model.3.rds',
file_refit = 'on_change'
)Step 3: posterior checks
summary(model.3)## Family: student
## Links: mu = identity; sigma = log; nu = identity
## Formula: change ~ condition + (1 | participant)
## sigma ~ condition
## Data: pose_df (Number of observations: 80)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Group-Level Effects:
## ~participant (Number of levels: 80)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 1.59 1.22 0.04 4.53 1.00 2896 1951
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept 24.56 5.03 14.96 34.83 1.00 5232
## sigma_Intercept 3.41 0.20 3.00 3.79 1.00 5833
## conditionexpansive -0.09 2.00 -4.06 3.84 1.00 10152
## sigma_conditionexpansive 0.15 0.22 -0.29 0.57 1.00 8612
## Tail_ESS
## Intercept 3332
## sigma_Intercept 3450
## conditionexpansive 2853
## sigma_conditionexpansive 2961
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## nu 5.98 7.52 1.70 25.11 1.00 5049 2838
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).cowplot::plot_grid(plot_predictions(model.3), p1, nrow = 2)
cowplot::plot_grid(plot_posteriors(model.3), p1, nrow = 2)
model.3.posteriors <-
model.3 %>%
epred_draws(tibble(condition = c('expansive', 'constrictive')),
re_formula = NA)wrap_plots(
model.3.posteriors %>%
mutate(model = 'model 3') %>%
rbind(
model.2.posteriors %>%
mutate(model = 'model 2')) %>%
rbind(
model.1.posteriors %>%
mutate(model = 'model 1')) %>%
ggplot() +
geom_density(aes(x = .epred, fill = condition), adjust = 1.5, color = NA, alpha = .5) +
facet_grid(model ~ .) +
scale_x_continuous(limits = c(-150, 250), breaks = seq(-150, 250, by = 50)) +
scale_color_theme() +
theme_density_x +
ggtitle('Compare the means of all three models'),
p1, nrow = 2, heights = c(4,1.5))
Model 4: Poisson / Negative-Binomial Regression model
Step 1: model specification
\[ \begin{align} y_{i} &\sim \mathrm{Neg-Binomial}(\mu, \phi) \\ \mu &= \beta_{i,j} + \beta_{1} * x_i \\ \beta_{1} &\sim \mathrm{Normal}(0, 1) \\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\}\\ j & \in \{1, ..., \mathrm{N}\} \end{align} \]
pose_raw_df = read.csv("data/posture_data-raw.csv") %>%
mutate(participant = factor(participant)) %>%
rename(trial = trial.number)
head(pose_raw_df)| participant | condition | total.money | trial | trial.money | exploded | pumps | life |
|---|---|---|---|---|---|---|---|
| 1 | expansive | 910 | 0 | 62 | 0 | 62 | 72 |
| 1 | expansive | 910 | 1 | 0 | 1 | 27 | 27 |
| 1 | expansive | 910 | 2 | 47 | 0 | 47 | 98 |
| 1 | expansive | 910 | 3 | 0 | 1 | 86 | 86 |
| 1 | expansive | 910 | 4 | 60 | 0 | 60 | 104 |
| 1 | expansive | 910 | 5 | 0 | 1 | 26 | 26 |
model.4 = brm(pumps ~ 1 + condition + trial + (trial | participant),
data = pose_raw_df, family = negbinomial,
prior = c(prior(normal(0, 10), class = Intercept),
prior(normal(0, 1), class = b),
prior(gamma(0.01, 0.01), class = shape)
), # this is the brms default
file = 'rds/model.4.rds',
file_refit = 'on_change' ,
iter = 4000, warmup = 1000, cores = 4, chains = 4)prior_summary(model.4)| prior | class | coef | group | resp | dpar | nlpar | lb | ub | source |
|---|---|---|---|---|---|---|---|---|---|
| normal(0, 1) | b | user | |||||||
| b | conditionexpansive | default | |||||||
| b | trial | default | |||||||
| normal(0, 10) | Intercept | user | |||||||
| lkj_corr_cholesky(1) | L | default | |||||||
| L | participant | default | |||||||
| student_t(3, 0, 2.5) | sd | 0 | default | ||||||
| sd | participant | default | |||||||
| sd | Intercept | participant | default | ||||||
| sd | trial | participant | default | ||||||
| gamma(0.01, 0.01) | shape | 0 | user |
summary(model.4)## Family: negbinomial
## Links: mu = log; shape = identity
## Formula: pumps ~ 1 + condition + trial + (trial | participant)
## Data: pose_raw_df (Number of observations: 2400)
## Draws: 4 chains, each with iter = 4000; warmup = 1000; thin = 1;
## total post-warmup draws = 12000
##
## Group-Level Effects:
## ~participant (Number of levels: 80)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(Intercept) 0.38 0.04 0.31 0.45 1.00 4207
## sd(trial) 0.01 0.00 0.00 0.01 1.00 3694
## cor(Intercept,trial) -0.62 0.12 -0.82 -0.35 1.00 9370
## Tail_ESS
## sd(Intercept) 6443
## sd(trial) 4265
## cor(Intercept,trial) 7921
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 3.52 0.06 3.41 3.63 1.00 2993 5184
## conditionexpansive -0.04 0.07 -0.18 0.10 1.00 3431 5201
## trial 0.01 0.00 0.00 0.01 1.00 8485 8646
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape 7.67 0.28 7.15 8.22 1.00 17805 8817
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Step 2: model interpretation
The results of this model are on the log-odds scale. What do the coefficients mean? The simplest way is to simply transform the data into a more interpretable scale and visualise the results:
# draws.model.4 =
wrap_plots(
plot_predictions(model.4, df = crossing(condition = c('expansive', 'constrictive'),
trial = 0:29,
participant = 0)),
p1,
nrow = 2)
We generate the average of 30 trials using an average participant.
model.4.posteriors <- model.4 %>%
epred_draws(crossing(condition = c('expansive', 'constrictive'),
trial = 0:29),
re_formula = NA,
allow_new_levels = FALSE) %>%
mutate(model = 'model 4') %>%
group_by(.draw, condition) %>%
summarise(.epred = mean(.epred))Plot out the posteriors for mean.
wrap_plots(
model.4.posteriors %>%
ggplot(aes(x = .epred, fill = condition)) +
geom_density(alpha = .5, size = 1, color = NA) +
scale_color_theme() +
facet_wrap(condition ~ ., ncol = 1) +
theme_density_x +
scale_x_continuous(limits = c(-100, 250), breaks = seq(-100, 250, by = 50)) +
ggtitle(paste0('Posterior means of ', deparse(substitute(model)))),
p1,
nrow = 2)
From this, we can see that there does not appear to be a difference between the two conditions.
Reporting
Let’s use model 4
model
We will need the ESSs and RHat from this print. Also, if you want to
report the standard deviation of random intercepts or CIs for other
paramaters. You can find them via summary(..)
summary(model.4)## Family: negbinomial
## Links: mu = log; shape = identity
## Formula: pumps ~ 1 + condition + trial + (trial | participant)
## Data: pose_raw_df (Number of observations: 2400)
## Draws: 4 chains, each with iter = 4000; warmup = 1000; thin = 1;
## total post-warmup draws = 12000
##
## Group-Level Effects:
## ~participant (Number of levels: 80)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(Intercept) 0.38 0.04 0.31 0.45 1.00 4207
## sd(trial) 0.01 0.00 0.00 0.01 1.00 3694
## cor(Intercept,trial) -0.62 0.12 -0.82 -0.35 1.00 9370
## Tail_ESS
## sd(Intercept) 6443
## sd(trial) 4265
## cor(Intercept,trial) 7921
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 3.52 0.06 3.41 3.63 1.00 2993 5184
## conditionexpansive -0.04 0.07 -0.18 0.10 1.00 3431 5201
## trial 0.01 0.00 0.00 0.01 1.00 8485 8646
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape 7.67 0.28 7.15 8.22 1.00 17805 8817
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Credible Intervals
We compute the credible intervals (CIs; Bayesian analogy to confidence intervals) for the two conditions.
model.4.posteriors.CI <-
model.4.posteriors %>%
group_by(condition) %>%
median_qi(.epred, width = .95)model.4.posteriors.CI| condition | .epred | .epred.lower | .epred.upper | width | width.lower | width.upper | .width | .point | .interval |
|---|---|---|---|---|---|---|---|---|---|
| constrictive | 36.84068 | 33.33245 | 40.68547 | 0.95 | 0.95 | 0.95 | 0.95 | median | qi |
| expansive | 35.44132 | 32.08602 | 39.12523 | 0.95 | 0.95 | 0.95 | 0.95 | median | qi |
model.4.posteriors %>%
ggplot() +
geom_density(aes(x = .epred, fill = condition), alpha = .5, color = NA) +
geom_point(model.4.posteriors.CI,
mapping = aes(x = .epred, y = 0), size = 3) +
geom_errorbarh(model.4.posteriors.CI,
mapping = aes(x = .epred, xmin = .epred.lower, xmax = .epred.upper, y = 0), height = 0, linewidth = 1.5) +
facet_wrap(condition ~ ., ncol = 1) +
scale_x_continuous(limits = c(0, 80)) +
scale_y_continuous(expand = c(.02,.02)) +
scale_color_theme() +
theme_density_x 
subtraction
model.4.posteriors_diff <-
model.4.posteriors %>%
ungroup() %>%
compare_levels(variable = .epred, by = condition) %>%
ungroup()head(model.4.posteriors_diff)| .draw | condition | .epred |
|---|---|---|
| 1 | expansive - constrictive | 0.4184726 |
| 2 | expansive - constrictive | 1.5736869 |
| 3 | expansive - constrictive | -5.8063069 |
| 4 | expansive - constrictive | -3.2075755 |
| 5 | expansive - constrictive | -2.7983548 |
| 6 | expansive - constrictive | -3.9121965 |
model.4.posteriors_diff.CI <-
model.4.posteriors_diff %>%
median_qi(.epred)
model.4.posteriors_diff.CI| .epred | .lower | .upper | .width | .point | .interval |
|---|---|---|---|---|---|
| -1.419472 | -6.401441 | 3.58453 | 0.95 | median | qi |
model.4.posteriors_diff %>%
ggplot() +
geom_density(aes(x = .epred), alpha = .5, fill = 'skyblue', color = NA, adjust = 2) +
geom_point(model.4.posteriors_diff.CI,
mapping = aes(x = .epred, y = 0), size = 3) +
geom_errorbarh(model.4.posteriors_diff.CI,
mapping = aes(x = .epred, xmin = .lower, xmax = .upper, y = 0), height = 0, linewidth = 1.5) +
#scale_x_continuous(limits = c(-50, 50)) +
xlab('expansive - constrictive') +
ggtitle('Mean difference in expansive and constrictive') +
geom_vline(xintercept = 0, linetype = 2) +
scale_y_continuous(expand = c(.02,.02)) +
scale_x_continuous(limits = c(-10, 10), breaks = seq(-10, 10, by = 5)) +
scale_color_theme() +
theme_density_x 
Session info
sessionInfo()## R version 4.2.2 (2022-10-31)
## Platform: aarch64-apple-darwin20 (64-bit)
## Running under: macOS Ventura 13.2
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] cmdstanr_0.5.2 knitr_1.39 ggplot2_3.4.0
## [4] bayesplot_1.9.0 ggdist_3.1.1 tidybayes_3.0.2
## [7] brms_2.17.0 Rcpp_1.0.8.3 modelr_0.1.8
## [10] broom.mixed_0.2.9.4 broom_1.0.0 patchwork_1.1.2
## [13] gtools_3.9.2.1 forcats_0.5.1 tidyr_1.2.0
## [16] purrr_0.3.4 tibble_3.1.7 dplyr_1.0.9
##
## loaded via a namespace (and not attached):
## [1] colorspace_2.0-3 ellipsis_0.3.2 ggridges_0.5.3
## [4] markdown_1.1 base64enc_0.1-3 rstudioapi_0.13
## [7] listenv_0.8.0 furrr_0.3.1 farver_2.1.1
## [10] rstan_2.21.5 bit64_4.0.5 svUnit_1.0.6
## [13] DT_0.23 fansi_1.0.3 mvtnorm_1.1-3
## [16] diffobj_0.3.5 bridgesampling_1.1-2 codetools_0.2-18
## [19] splines_4.2.2 shinythemes_1.2.0 jsonlite_1.8.0
## [22] shiny_1.7.1 readr_2.1.2 compiler_4.2.2
## [25] backports_1.4.1 assertthat_0.2.1 Matrix_1.5-1
## [28] fastmap_1.1.0 cli_3.4.1 later_1.3.0
## [31] htmltools_0.5.2 prettyunits_1.1.1 tools_4.2.2
## [34] igraph_1.3.1 coda_0.19-4 gtable_0.3.0
## [37] glue_1.6.2 reshape2_1.4.4 posterior_1.2.1
## [40] jquerylib_0.1.4 vctrs_0.5.1 nlme_3.1-160
## [43] crosstalk_1.2.0 tensorA_0.36.2 xfun_0.31
## [46] stringr_1.4.0 globals_0.15.0 ps_1.7.0
## [49] mime_0.12 miniUI_0.1.1.1 lifecycle_1.0.3
## [52] future_1.26.1 zoo_1.8-10 scales_1.2.0
## [55] vroom_1.5.7 colourpicker_1.1.1 hms_1.1.1
## [58] promises_1.2.0.1 Brobdingnag_1.2-7 parallel_4.2.2
## [61] inline_0.3.19 shinystan_2.6.0 yaml_2.3.5
## [64] gridExtra_2.3 loo_2.5.1 StanHeaders_2.21.0-7
## [67] sass_0.4.1 stringi_1.7.6 highr_0.9
## [70] dygraphs_1.1.1.6 checkmate_2.1.0 pkgbuild_1.3.1
## [73] rlang_1.0.6 pkgconfig_2.0.3 matrixStats_0.62.0
## [76] distributional_0.3.0 evaluate_0.15 lattice_0.20-45
## [79] labeling_0.4.2 rstantools_2.2.0 htmlwidgets_1.5.4
## [82] cowplot_1.1.1 bit_4.0.4 tidyselect_1.1.2
## [85] processx_3.5.3 parallelly_1.31.1 plyr_1.8.7
## [88] magrittr_2.0.3 R6_2.5.1 generics_0.1.3
## [91] DBI_1.1.2 pillar_1.7.0 withr_2.5.0
## [94] xts_0.12.1 abind_1.4-5 crayon_1.5.1
## [97] arrayhelpers_1.1-0 utf8_1.2.2 tzdb_0.3.0
## [100] rmarkdown_2.14 grid_4.2.2 callr_3.7.0
## [103] threejs_0.3.3 digest_0.6.29 xtable_1.8-4
## [106] httpuv_1.6.5 RcppParallel_5.1.5 stats4_4.2.2
## [109] munsell_0.5.0 bslib_0.3.1 shinyjs_2.1.0